# Closed form for $\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...}}}$

Is there a close form for the following nested radical?

$$\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...}}}$$

It converges and
$$\quad \quad \lim_{n \to\infty} \sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...+\sqrt[n!]{n^n}}}}=1.8430759846682...$$

Is this number algebraic or transcendental?

• If it has a closed form, then $e$ is likely involved. Jul 2, 2015 at 21:21
• I honestly doubt that a closed form will be found, but of course you never know. Jul 2, 2015 at 21:30
• Is there any context from which the above constant arises? If this is just a radical made on a spot, I see no reason for it to have a closed form. Jul 2, 2015 at 21:39
• I conjecture that almost surely 1) it does not have a closed form 2) it is trancendental 3) nobody will be able to provide an answer (proof) in one way or another to either of the two questions. Jul 2, 2015 at 21:49

• @QuantumDot: Never mind, I forgot to extract the square root. My sequence started $\sqrt[1!]{1^1+\sqrt[2!]{2^2+\ldots}}$ Nov 27, 2015 at 10:44